One Way ANOVA ©2005 Dr. B. C. Paul modified 2009 Note – The concepts presented in these slides are considered common knowledge to those familiar with statistics and similar ideas are found in many texts. The approach to the topic presented here was chosen by the author and is not known to follow outlines covered in other books. The slides do contain screen shots taken from the output of the program SPSS.

Comparing Populations We last considered a case where we compared two populations to determine if they were the same We last considered a case where we compared two populations to determine if they were the same The Red Rooster Carburetor The Red Rooster Carburetor Sometimes a particular variable involves many cases and populations – going two by two might take longer than Noah loading the arc Sometimes a particular variable involves many cases and populations – going two by two might take longer than Noah loading the arc

Consider the Case of the Red Rooster Carburetor We compared an array of cars with and without the carburetor to see if it improved mileage We compared an array of cars with and without the carburetor to see if it improved mileage It did but we needed a huge set of trials It did but we needed a huge set of trials We picked one particular type of car and went pair-wise to see if the carburetor improved mileage on one particular type of car and over a wide range of drivers We picked one particular type of car and went pair-wise to see if the carburetor improved mileage on one particular type of car and over a wide range of drivers We needed fewer trials but we only knew about 1 type of car in the end. We needed fewer trials but we only knew about 1 type of car in the end.

Suppose Suppose we want to know whether the Red Rooster Carburetor improves mileage in a wide range of cars. Suppose we want to know whether the Red Rooster Carburetor improves mileage in a wide range of cars. Right now we only know it works on average of a bunch of cars and drivers and that it probably works on Dodge Neons with a wide range of different drivers Right now we only know it works on average of a bunch of cars and drivers and that it probably works on Dodge Neons with a wide range of different drivers

Enter ANOVA Ok What is ANOVA Ok What is ANOVA ANOVA standards for Analysis of Variance ANOVA standards for Analysis of Variance The Technique puts test data together in groups obtained under like conditions The Technique puts test data together in groups obtained under like conditions It then looks at differences and variability's between different groups It then looks at differences and variability's between different groups The technique tells us which changes in conditions are producing abnormal changes in results The technique tells us which changes in conditions are producing abnormal changes in results Those changes will then be considered statistically significant Those changes will then be considered statistically significant

The Experiment We will repeat the Dodge Neon experiment with several different types of cars We will repeat the Dodge Neon experiment with several different types of cars What we want to know is whether the results we get from the Red Rooster Carburetor are different for different types of cars What we want to know is whether the results we get from the Red Rooster Carburetor are different for different types of cars

The Data For a Dodge Neon where we had 10 different drivers drive the car before and after changing the Carburetor we found For a Dodge Neon where we had 10 different drivers drive the car before and after changing the Carburetor we found Results 21.13%, 17.12%, 26.19%, 24.68%, 21.71%, 16.79%, 20.48%, 22.51%, 22.85%, 22.34% Results 21.13%, 17.12%, 26.19%, 24.68%, 21.71%, 16.79%, 20.48%, 22.51%, 22.85%, 22.34% This represents improvement in gas mileage This represents improvement in gas mileage

More Test Work We had the same 10 drivers do before and after drives in a Ford E350 Van, a Cadillac Deville, a Honda Accord, a Chevy Malibu, and a Toyota Tundra We had the same 10 drivers do before and after drives in a Ford E350 Van, a Cadillac Deville, a Honda Accord, a Chevy Malibu, and a Toyota Tundra We calculated our % improvement in fuel economy for each driver in each car We calculated our % improvement in fuel economy for each driver in each car Our Question – does the improvement depend on the type of car that was retrofit with the Red Rooster Carburetor? Our Question – does the improvement depend on the type of car that was retrofit with the Red Rooster Carburetor?

Setting Up ANOVA Dodge Neons Data Ford E350 Van Data Cadillac Deville Data Honda Accord Data Chevy Malibu Data Toyota Tundra Data We group our gas mileage improvements by type of car

Ideas About ANOVA If we take our data and square it – big numbers will have big squares and little numbers little squares If we take our data and square it – big numbers will have big squares and little numbers little squares All the numbers squared is our sum of squares total All the numbers squared is our sum of squares total We can get subtotal sums of squares within our individual “Treatments” We can get subtotal sums of squares within our individual “Treatments” Anything not accounted for by the treatments must be due to random error Anything not accounted for by the treatments must be due to random error

Lets Do the Problem I’m not going to show you how to calculate the sums of squares I’m not going to show you how to calculate the sums of squares I’m not going to tell you much about the mathematical theory of why we are doing this I’m not going to tell you much about the mathematical theory of why we are doing this I will show you how to set this up and let SPSS run the analysis I will show you how to set this up and let SPSS run the analysis

Set #1 Enter the Data in SPSS Note that I entered before and After fuel economy improvements For each driver in column #1 I entered a numeric code for auto Type in field #2 where 1 was a Neon 2 is a Ford E350, 3 is a Cadillac DeVille etc.

Step #2 Tell SPSS to do a One Way ANOVA Pull down the menu under Analyze Highlight Compare Means On the pop out menu highlight One Way Anova and hit return

Make Improvement in Gas Mileage the “Dependent Variable” Highlight Improvement In the box on the left Click the arrow by Dependent Variable to Move Improvement to The dependent variable Box.

Make Auto the Factor Highlight Auto Click on the arrow to move the variable Into the factor box.

Click OK and Read the Results

Understanding Results First Thing ANOVA did was to compile the squares Of the values in the different groupings. We see of the total variability most of the Differences are showing up inside the groups With very little difference between groups

Review the Degrees of Freedom We have 6 types of cars With n-1 degrees of Freedom or 5. We have 60 pieces of data to analyze so we have n-1 total Degrees of Freedom. Of the 59 total degrees of freedom we use up 5 of them on Different types of cards leaving 54 values free to send results In random directions.

We get Our Mean Square by Divide the SS by the Degrees of Freedom Our Mean Square This is giving us an idea of the average amount of variability Coming from something – in this case variability by type of car And variability coming from everything else.

We next divide the mean square for the treatment by the mean square for error (everything else) Result of our Division So what is this F business?

I Was Afraid You Were Going to Tell Me If our model fits the real world when we divide our normally distributed mean square for treatments by our normally distributed mean square error the resulting number has an F distribution. If our model fits the real world when we divide our normally distributed mean square for treatments by our normally distributed mean square error the resulting number has an F distribution. F distribution is just another of those probability density functions that Sadisticians – woops Statisticians worked out proved and integrated and put results in tables. F distribution is just another of those probability density functions that Sadisticians – woops Statisticians worked out proved and integrated and put results in tables.

As with Other Tests Value we calculated goes on the X axis. Integrate the area under the curve up to the value of X. This Represents the chances of getting a less unusual result. The area above that point represents The risk you take if you reject the “Null Hypothesis”

So What is the “Null Hypothesis” this time The “Null Hypothesis” is the assumption that nothing beyond unaccounted for random variations are taking place. The “Null Hypothesis” is the assumption that nothing beyond unaccounted for random variations are taking place. In the case of a one way ANOVA the “Null Hypothesis” is that our treatment variable makes no difference In the case of a one way ANOVA the “Null Hypothesis” is that our treatment variable makes no difference We will look for our test statistic to have such a high “degree of significance” that we just can’t believe that We will look for our test statistic to have such a high “degree of significance” that we just can’t believe that

Looking at Our Significance The significance shown here is the area under the curve Beyond our point on the X axis. One basically means that the whole Universe of opportunities is Still out there – Ie The type of Car being fit with the Red Rooster Carburetor makes no difference compared to other Factors that have not yet been explained.

Practical Meaning We cannot reject the “Null Hypothesis” that type of car does not influence the results of adding a Red Rooster Carburetor. We cannot reject the “Null Hypothesis” that type of car does not influence the results of adding a Red Rooster Carburetor. If we were going to market this Carburetor this would mean that the carburetor is likely to give anyone the improvement in fuel economy regardless of what kind of car they drive If we were going to market this Carburetor this would mean that the carburetor is likely to give anyone the improvement in fuel economy regardless of what kind of car they drive Not rejecting the “Null Hypothesis” is not always bad. Not rejecting the “Null Hypothesis” is not always bad.

Limitations in Meaning We would like to jump to judgment and say “the type of car makes no difference”. We would like to jump to judgment and say “the type of car makes no difference”. Two numbers influence the F tests. Two numbers influence the F tests. Big F numbers can come from big MS effects for treatments Big F numbers can come from big MS effects for treatments They can also come from little denominator values for MS error They can also come from little denominator values for MS error

Mean Square Error Mean Square Error is really just the variability from everything else that you did not consider in the test. Mean Square Error is really just the variability from everything else that you did not consider in the test. If MS error is big it can make an important treatment effect seem to be insignificant by comparison If MS error is big it can make an important treatment effect seem to be insignificant by comparison It can actually mean you need to do more work to figure out what you are missing before you can analyze anything It can actually mean you need to do more work to figure out what you are missing before you can analyze anything

Assumptions and Limitations We assume we are studying normally distributed populations We assume we are studying normally distributed populations We assume that the variance is the same (or negligibly close) for each of our treatments We assume that the variance is the same (or negligibly close) for each of our treatments Ie the variability in results for gas mileage improvement are the same regardless of which type of car was tested Ie the variability in results for gas mileage improvement are the same regardless of which type of car was tested The math requires us to have the same number tests for each different type of car The math requires us to have the same number tests for each different type of car There are specialized test arrangements that get you around some of these standard limiting provisions. There are specialized test arrangements that get you around some of these standard limiting provisions.

Now It’s Your Turn Do Unit #3 Assignment #1 (also known as assignment #5) Do Unit #3 Assignment #1 (also known as assignment #5) When you build a highway you want the pavement to take a long time before it breaks up When you build a highway you want the pavement to take a long time before it breaks up People are also looking for ways to make pavement mixes stonger, longer lasting and cheaper People are also looking for ways to make pavement mixes stonger, longer lasting and cheaper One possible way is to add Fly Ash and replace cement One possible way is to add Fly Ash and replace cement When coal is burned in a power plant it contains some dirt that settled with the plant material in the swamp When coal is burned in a power plant it contains some dirt that settled with the plant material in the swamp Dirt does not burn but the little dirt particles melt and are carried up the smokestack by the hot moving air Dirt does not burn but the little dirt particles melt and are carried up the smokestack by the hot moving air We put clears on the smokestacks so we don’t dump soot all over the country side We put clears on the smokestacks so we don’t dump soot all over the country side The stuff we collect is a gray or tan colored talcum powder like texture – its called fly ash. The stuff we collect is a gray or tan colored talcum powder like texture – its called fly ash. Different amounts of fly ash are added to road pavement mixes. The question is whether it changes how long the road lasts. Different amounts of fly ash are added to road pavement mixes. The question is whether it changes how long the road lasts.